Lower and upper local uniform K-monotonicity in symmetric spaces

Abstract

Using the local approach to the global structure of a symmetric space $E$, we establish a relationship between strict $K$-monotonicity, lower (resp., upper) local uniform $K$-monotonicity, order continuity, and the Kadec–Klee property for global convergence in measure. We also answer the question: Under which condition does upper local uniform $K$-monotonicity coincide with upper local uniform monotonicity? Finally, we present a correlation between $K$-order continuity and lower local uniform $K$-monotonicity in a symmetric space $E$ under some additional assumptions on $E$.